It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acting on the lines is a subgroup of the Weyl group $W(E_6)$ which has order $51840$.

I was wondering if a similar argument can prove the existence of an absolute constant $c$ such that whenever a smooth cubic surface has coefficients in a number field $K$ then all $27$ lines are defined in a number field extension $L$ of $K$ of degree at most $[L:K]\leq c.$